Permutation and Multi-permutation Codes Correcting Multiple Deletions

TL;DR

This paper advances permutation codes in the Ulam metric by improving bounds, designing efficient decoding algorithms for multiple deletions, and establishing new mappings connecting permutation codes across different metrics.

Contribution

It introduces a new mapping linking permutation codes in Hamming and Ulam metrics, improves bounds on code size, and constructs codes correcting multiple deletions and bursts with efficient decoding.

Findings

Improved Gilbert--Varshamov bound for permutation codes

Designed permutation codes correcting multiple deletions with low redundancy

Constructed best-known multi-permutation codes in Ulam metric

Abstract

Permutation codes in the Ulam metric, which can correct multiple deletions, have been investigated extensively recently. In this work, we are interested in the maximum size of permutation codes in the Ulam metric and aim to design permutation codes that can correct multiple deletions with efficient decoding algorithms. We first present an improvement on the Gilbert--Varshamov bound of the maximum size of these permutation codes by analyzing the independence number of the auxiliary graph. The idea is widely used in various cases and our contribution in this section is enumerating the number of triangles in the auxiliary graph and showing that it is small enough. Next, we design permutation codes correcting multiple deletions with a decoding algorithm. In particular, the constructed permutation codes can correct $t$ deletions with at most $(3t-1) \log n+o(\log n)$ bits of redundancy where $n$ is the length of the code. Our construction is based on a new mapping which yields a new connection between permutation codes in the Hamming metric and permutation codes in various metrics. Furthermore, we construct permutation codes that correct multiple bursts of deletions using this new mapping. Finally, we extend the new mapping for multi-permutations and construct the best-known multi-permutation codes in Ulam metric.